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Quantum computing and computational law
Jeffery Atik & Valentin Jeutner
To cite this article: Jeffery Atik & Valentin Jeutner (2021): Quantum computing and computational
law, Law, Innovation and Technology, DOI: 10.1080/17579961.2021.1977216
To link to this article: https://doi.org/10.1080/17579961.2021.1977216
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Quantum computing and computational law
and Valentin Jeutner
Loyola Law School, Loyola Marymount University, Los Angeles, CA, USA;
Faculty of Law,
Lund University, Lund, Sweden
Quantum computing technology will greatly enhance the abilities of the
emerging ﬁeld of computational law to express, model, and operationalise
law in algorithmic form. Foreshadowing the harnessing of the power of
quantum computing technology by the legal sector, this essay targets, with
reference to computational complexity theory, the categories of
computational problems which quantum computers are better equipped to
deal with than are classical computers (‘quantum supremacy’).
Subsequently, the essay demarcates the possible contours of legal
‘quantum supremacy’by showcasing three anticipated legal ﬁelds of
quantum technology: optimisation problems, burdens of proof, and
machine learning. Acknowledging that the exact manifestation of quantum
computing technology in the legal sector is as yet diﬃcult to predict, the
essay posits that the meaningful utilisation of quantum computing
technology at a later stage presupposes a creative imagination of possible
use-cases at the present.
ARTICLE HISTORY Received 19 October 2020; Accepted 21 March 2021
KEYWORDS Quantum law; quantum computer; quantum mechanics; computational law; innovation
In 1998, John Preskill asked two fundamental questions concerning
quantum computers: do we want to build them, and can we build
Twenty years later, as the ﬁrst quantum computers are being con-
structed, these questions have been answered in the aﬃrmative and the
identiﬁcation of use cases for quantum computing is in full swing.
© 2021 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group
This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDer-
ivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distri-
bution, and reproduction in any medium, provided the original work is properly cited, and is not altered,
transformed, or built upon in any way.
CONTACT Jeﬀery Atik jeﬀery.email@example.com Burns 343, 919 Albany St., Los Angeles, CA
90015, USA; Valentin Jeutner firstname.lastname@example.org Faculty of Law, University of Lund, 22100
John Preskill, ‘Quantum Computing: Pro and Con’(1998) 454 Proceedings of the Royal Society of London
LAW, INNOVATION AND TECHNOLOGY
the next few years, both the European Union
and the United States
committed to set aside 1 billion Euros and 1.2 billion USD, respectively, to
support quantum research. Meanwhile, car manufacturers are teaming up
with technology companies to enhance the eﬃcacy of autonomous
systems (e.g. VW and Google,
Daimler and IBM Q
) while pharma-
ceutical ﬁrms aim to utilise quantum computers to dramatically enhance
the abilities of AI-supported pattern-recognition in the interest of more
precise patient diagnostics.
These initiatives are not only driven by the
expectation that quantum computers are able to perform tasks currently
performed by classical computers more eﬃciently (‘quantum advantage’),
but also anticipate that quantum computers will be able to perform
tasks that classical computers cannot perform at all (‘quantum
Against this background, this essay asks: what, if anything, will quantum
computing bring to law?
Speciﬁcally, this essay considers the potential of
quantum computing to facilitate the emergence of computational law –the
expression, application and analysis of law in algorithmic form.
The challenge today, in peering forward into the age of quantum compu-
ters, is to imagine legal applications that can actually harness ‘quantum
supremacy’since there is little or no reason (beyond exercising technological
competencies) to task quantum computers with solving problems that clas-
sical computers can already successfully deal with. Thus, it is one of the aims
Francesco Bova, Avi Goldfarb and Roger G Melko, ‘Commercial Applications of Quantum Computing’
(2021) 8 EPJ Quantum Technology 1.
High Performance Computing and Quantum Technology Unit, ‘Quantum Technologies Flagship’(Digital
Single Market - European Commission, 12 May 2016) <https://ec.europa.eu/digital-single-market/en/
quantum-technologies> accessed 21 June 2021.
Martin Giles, President Trump has signed a $1.2 billon law to boost US quantum tech,MIT Technology
boost-us-quantum-tech/ (last visited Aug 4, 2019).
Volkswagen, ‘Volkswagen Group and Google Work Together on Quantum Computers’(7 November
2017) <media.vw.com> accessed 21 June 2021.
Svenja Gelowicz, ‘IBM und Daimler entwickeln Quantencomputer’(14 November 2017) <https://www.
automobil-industrie.vogel.de/ibm-und-daimler-entwickeln-quantencomputer-a-671718/> accessed 21
Esther O’Sullivan, ‘Quantum Computing, Artiﬁcial Intelligence and Health Care’(BMJ Technology Blog,3
November 2017) <https://blogs.bmj.com/> accessed 21 June 2021.
According to John Preskill ‘quantum supremacy’exists when ‘controlled quantum systems’can perform
tasks that ‘ordinary digital computers’cannot. Sometimes the terms ‘quantum supremacy’and
‘quantum advantage’are used interchangeably. John Preskill, ‘Quantum Computing and the Entangle-
ment Frontier’ Proceedings of the 25th Solvay Conference on Physics 1, 2.
There is a rich literature involving quantum theory to stimulate thinking about law. For examples, see
Ted Sichelman, ‘Quantifying Legal Entropy’(2021) 9 Frontiers in Physics; William HJ Hubbard, ‘Quantum
Economics, Newtonian Economis, and Law’ Michigan State Law Review 425; Henry E Smith,
‘Modularity in Contracts: Boilerplate and Information Flow’(2006) 104 Michigan Law Review 1175.
For a comprehensive overview of computational law, see Kevin D. Ashley, Artiﬁcial Intelligence and
Legal Analytics - New Tools for Law Practice in the Digital Age (Cambridge University Press, 2017); Natha-
niel Love and Michael Genesereth, ‘Computational Law’ Proceedings of the 10th International
Conference on Artiﬁcial Intelligence and Law 205.
2J. ATIK AND V. JEUTNER
of this essay to address this challenge by explaining which kind of problems
might be able to harness ‘quantum supremacy’.
When attempting to identify instances of ‘quantum supremacy’, compu-
tational complexity theory is a helpful point of departure. Computational
complexity theory describes a given problem in terms of the classical com-
puting resources required to solve it.
All problems (other than those
with constant solutions) increase in complexity as the number of factors
increases. Problems can be classiﬁed into a hierarchy of complexity cat-
egories. Simpler problems are linear in nature: the amount of demanded
resources increases in a linear relationship to the number of factors. These
are easily managed by classical computers. If we ask a computer to output
the number of boots required by an army, the problem does not become
meaningfully more diﬃcult as the number of soldiers increases from
100,000–10,000,000; the number of needed boots increases in a linear pro-
portion to the number of soldiers (that is, two times the number of soldiers).
The computer may require a bit more time to calculate the required number
of boots as the number of soldiers increases, but the required computational
energy should not increase by more than a ﬁxed proportion.
More diﬃcult problems require polynomial resources. This category is
known as P, where there is a known, eﬃcient algorithm to determine a sol-
ution. Here complexity increases as some power of the number of factors.
While more diﬃcult (in the sense that these problems demand increasingly
greater computer resources), these problems can also be solved (assuming a
reasonable upper bound on the number of factors).
By the time a problem reaches 20 factors, the complexity for certain
P-type problems (even when a solution demonstrably exists) would
require more computer resources (processing, energy and time) than is avail-
able in a near-eternity. These tough problems are provisionally grouped in a
subset known as NP (the ‘N’suggests that these problems are ‘non-determi-
nistic’–which eﬀectively means there is no known, practical algorithm that
can shorten the pathway to an eﬃcient solution). If we are given a solution to
an NP problem, we can verify that it is correct; however, we cannot solve an
NP problem (not knowing the solution) in polynomial time.
The most diﬃcult problems have exponential solutions (such problems
belong to the complexity category known as EXP). Here the complexity of
the problem increases in a power that depends on the number of factors.
For a general overview, see Walter Dean, ‘Computational Complexity Theory’in Edward N Zalta (ed),
Stanford Encyclopedia of Philosophy (2016) <https://plato.stanford.edu/archives/win2016/entries/
computational-complexity> accessed 21 June 2021.
‘Pis the class of problems solvable by a Turing machine in polynomial time. In other words, Pis the
union, over all positive integers k, of TIME(n
).’Scott Aaronson, Quantum Computing since Democritus
(Cambridge University Press, 2013) 55.
LAW, INNOVATION AND TECHNOLOGY 3
As the number of factors increases from 1 to 2, the complexity is squared. As
the number moves from 2 to 3, the measure of complexity is cubed.
Computational complexity theory is a good starting point to identify pro-
blems for which quantum computers might demonstrate ‘quantum supre-
macy’. However, simply because a problem is too complex for a classical
computer, we cannot presume that quantum computers will perform any
better. Computational complexity theory –for the moment –is itself pre-
mised on the capabilities of classical computers; it will inevitably expand
to include additional categories of problems by their relative diﬃculty to
solve using quantum computers.
Until mathematicians identify precisely
which algorithms are tractable using quantum computers, the parameters
of ‘quantum supremacy’will remain unclear.
With respect to law, there is the added uncertainty concerning the extent
to which computational complexity theory could be used to assess legal
Computational law is in its infancy. Most stated legal algor-
ithms (even those deemed by lawyers to be complex) are not complex in a
computational (computer resource) sense.
But just as new and more
powerful algorithms emerge, computational law, too, will develop. The pro-
mulgation of law in computational form
and the accelerating translation of
law from human language to computer code will facilitate the advent of
complex quantum legal algorithms.
In light of these inevitable uncertainties, we outline in this essay several
categories of problems where anticipated ‘quantum supremacy’could be of
legal relevance. One of those categories concerns optimisation problems.
The law is replete with multi-factor tests, where judges or legislatures
create lists of factors that are to be considered (as a matter of process) or
weighed (as in ‘balancing tests’) before reaching a legal conclusion. These
multi-factor mandates may be translated into complex algorithms that fall
into the complexity class NP. This means that they will not be solvable
using classical computers. These legal optimisation challenges may,
however, be resolved using quantum computers (assuming appropriate
quantum algorithms can be identiﬁed for them). A second category of pro-
blems relates to law’s elaborate rules for allocating and resolving questions of
burden of proof on an operational level. Computational law will have to
properly model the operations of these rules to have the requisite power.
Burdens of proof necessarily involve the simultaneous operation of many
Aaronson (n 12) presents the quantum complexity class BQP, the class of problems solvable on a
quantum computer in polynomial time. ibid 136.
Eric Kades ﬁrst assessed the relevance of computational complexity theory to law in a 1997 article. Eric
Kades, ‘The Laws of Complexity and the Complexity of Laws: The Implications of Computational Com-
plexity Theory for Law’(1997) 49 Rutgers Law Review 403.
See the discussion of various juridical notions of complexity by Kades (n 14).
Already, US legislation must be expressed in machine-readable form. Open Government Data Act of
2019, S 760 / HR 1770.
4J. ATIK AND V. JEUTNER
factors. As such, administration of burdens of proof may be an area of com-
putational law well-suited for quantum computing. Finally, the utilisation of
quantum computing based machine learning may enhance the ability to
create legal models.
This essay addresses the prospects of these three areas for further explora-
tion. However, in order to contextualise our discussion in the essay’sﬁnal
part, we commence with an introduction of the promise of quantum com-
puting in section 2and unearth the speciﬁcally legal implications of
quantum computing for computational law in section 3.
2. The promise of quantum computing
An appreciation of the legal signiﬁcance of quantum computing presupposes
an accurate understanding of the characteristics of quantum computing
technology. Thus, this ﬁrst section of the essay brieﬂy explains the function
of quantum computers (2.1), introduces the ﬁrst identiﬁed quantum algor-
ithm (2.2) and relates quantum technology to computational complexity
2.1. The power of qubits
Already, operationalised algorithms are used in the emerging ﬁeld of legal
analytics to model, predict or instantiate the legal system. Inevitably, law
will move –at least in part –to computer code, in a transition that resembles
law’s movement from orality to text.
The promised range of AI-related
technologies will supplement human operators of the legal system. In this
regard, quantum computing opens up speciﬁc potentialities for compu-
tational law, enabling new (and possibly strange
However, in order to address the question of what quantum computing
can and will bring to law, when quantum computing arrives as a deployed
technology, we must ﬁrst explore the more general question of what
quantum computing can achieve that is beyond the reach of classical
Ordinarily, a classical computer stores and processes information in
binary units, called bits. These bits can have the value of either 1 or
For a treatment of the signiﬁcance of law’s transition from orality to written codes, see Peter M Tiersma,
Parchment, Paper, Pixels: Law and the Technologies of Communication (University of Chicago Press,
2010). See also Rosalind Thomas, ‘Written in Stone? Liberty, Equality, Orality and the Codiﬁcation of
Law’(1995) 40 Bulletin of the Institute of Classical Studies 59.
Quantum phenomena, such as wave/particle duality, are often described as ‘strange’or ‘weird’. These
phenomena display characteristics or behaviors that do not correspond to ordinary human sense
experience. A ‘strange’quantum algorithm might follow a surprising pathway to reach its result.
See, for example, Daniel F Styer, The Strange World of Quantum Mechanics (Cambridge University
LAW, INNOVATION AND TECHNOLOGY 5
0. Quantum computers do not use bits, but rather qubits, to store and
process information. Qubits can be set to 1 or 0, like a classic computer.
But, importantly, they can also be set to be 1 and 0 at the same time. This
technological diﬀerence is the reason that quantum computers are vastly
more powerful than classical computers. To illustrate, one of the manufac-
turers of quantum computers recently reported that their quantum computer
performed a calculation in 1 s that would take a classic computer 10 000 years
In general, it is anticipated that fully functioning quantum
computers will be 100 million times more powerful than contemporary
desktop computers and at least 3500 times more powerful than contempor-
ary super-computers. Since quantum computers operate through the
manipulation of an actual quantum system it is no surprise that they can
be usefully deployed to model other quantum systems. For example, the
folding of complex biomolecules may involve quantum mechanical pro-
cesses; and so quantum computers may facilitate the modelling of these pro-
cesses, permitting designer biomolecules.
It is important to appreciate at this stage that, with respect to certain pro-
blems, the potential speed of quantum computers is so superior to that of
classical computers that problems hitherto intractable by classical computers
become tractable. That does not mean, however, that quantum computers
will replace classical computers across the board. Indeed, quantum compu-
ters are not per se faster or more powerful than classical computers. Both
types of computer will co-exist, each with strengths in its own domain.
Indeed, there appear to be applications where hybrid computer systems,
involving both quantum and classical processors, may be optimum.
sequently, much of quantum computing is identifying which problems fall
within the class of problems that quantum computing can easily crack.
To see the possibilities for law, we have to ask whether there are legal
questions that are suited to quantum computers. There is the intriguing
possibility that law, or at least certain aspects of law, may be revealed to
have a fundamental quantised structure
–much as Newton’s mechanics
was revealed to be only a generalisation of more speciﬁc quantum mechanics
when taken to a large scale.
Mark Molloy, ‘Google’s New Quantum Computer Is “100 Million Times Faster than Your Pc”’ The Tele-
graph (9 December 2015) <https://www.telegraph.co.uk> accessed 21 June 2021.
See Carlos Outeiral and others, ‘The Prospects of Quantum Computing in Computational Molecular
Biology’(2021) 11 WIREs Computational Molecular Science.
See Bova, Goldfarb and Melko (n 4).
For a legal invocation of quantum mechanical concepts, see Valentin Jeutner, Irresolvable Norm
Conﬂicts in International Law: The Concept of a Legal Dilemma (Oxford University Press, 2017) 3. See
also Tamara Ćapeta, ‘Do Judicial Decision-Making and Quantum Mechanics Have Anything in
Common? A Contribution to Realist Theories of Adjudication at the CJEU’in Martin Belov (ed), The
Role of Courts in Contemporary Legal Orders (Eleven, 2019).
Intriguingly, the 1787 Constitution of the United States has, for example, been described as ‘Newtonian
in design, with its carefully counterpoised forces and counterforces, its checks and balances’. Lawrence
6J. ATIK AND V. JEUTNER
We can imagine at least one set of problems that is relevant to law where
quantum computing holds promise: complex optimisation problems. What
appear to ordinary mortals to be discrete problems, mathematicians regu-
larly see as one and the same problem.
Many optimisation problems are
understood to be versions of the travelling salesman’s problem (also
known as the TSP)
which involves optimising a pathway linking a large
set of dispersed points. Optimal designs of computer chips are solutions to
Imagine a travelling salesman needs to visit four cities, which are separ-
ated by various distances. Given a small number of cities (and knowing
the distances between each possible city pair), one can readily determine
the shortest pathway linking those four cities. Here one seeks to optimise
the travelling salesman’s trip by shortening it.
While the TSP can be solved quite easily for a small number of cities, it
quickly becomes intractable, as the combinations of pathways among, say
26 cities, is astronomical. This is a form of optimisation problem that
would take a classical computer ages to solve. And it is one that, at least in
theory, a quantum computer could solve in seconds.
For the moment, we do not have a general view of law as a quantum
system. Indeed, it likely is not even a classical system; ‘legal logic’only
vaguely correlates to classical logic, and there is much to law that deﬁes clas-
sical modelling. But that does not mean that a day might not come where law
is recognised to be fundamentally quantised. And if that is the case, then
quantum computers would be particularly suited to model, predict and
instantiate law. A prominent point of departure for imagining the compe-
tences of quantum computers is Shor’s Algorithm which will be explored
in the next section.
2.2. Shor’s algorithm
There are at present few identiﬁed applications where quantum computers
have a signiﬁcant advantage to classical computers. This does not mean
that there may not ultimately be many more problems identiﬁed in the
future where quantum computing will enjoy an advantage. One application
where quantum computers dominate classical computers is large number
H Tribe, ‘The Curvature of Constitutional Space: What Lawyers can Learn from Modern Physics’(1989)
103 Harvard Law Review 1, 3.
‘We can say that almost all of the ‘hard’problems are the same hard problem in diﬀerent guises –in
the sense that, if we had a polynomial-time algorithm for any one of them, then we’d also have poly-
nomial-time algorithms for all the rest.’Aaronson (n 14) 57.
See William J Cook, In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation (Prin-
ceton University Press, 2012).
LAW, INNOVATION AND TECHNOLOGY 7
Large number prime factorisation involves the identiﬁcation of the two
prime numbers that, when multiplied, generate the large number of interest.
Large number prime factorisation is well-recognised as a diﬃcult problem in
number theory. There is no known (non-quantum) algorithm available for
identifying the two prime factors that, when multiplied, give the deﬁned
large number product. In other words, there is no known direct path to iden-
tifying the factors if we have their product. Rather, one has to use slow and
laborious techniques, just slightly more eﬀective than trial and error.
Large number prime factorisation is a nice example of so-called one-way
in mathematics. Given two primes, it is very easy to ﬁnd their
product. For example, 4507 and 7883 are both prime numbers. A schoolchild
can, through a small number of operations, compute their product:
35,528,681. A computer can reach the same product (performing the algor-
ithm for multiplication of two integers) in a fraction of a second. But things
are much more diﬃcult working in the opposite direction. It would take a lot
of trial and error to ﬁnd the prime factors of 35,528,681 if you did not already
know them –more challenging for the human as well as the computer, as
both lack a tractable algorithm that can be applied to the problem.
Utilisation of one-way functions, such as large number prime factoris-
ation, underlie much of contemporary cryptography. It is easy to ﬁnd the
result working in one direction (determining the product, given
the factors) and very diﬃcult to do so, working in the other (determining
the factors, given the product).
This well-known, very diﬃcult problem (the inherent diﬃculty of which
contributes to its usefulness) will likely yield to quantum computing. Shor’s
is the ﬁrst important quantum algorithm identiﬁed that, when
implemented on a viable quantum computer, will elegantly and eﬀectively
solve the problem of large number prime factorisation. And again, obtaining
solutions to this problem is not devoid of practical signiﬁcance: a quantum
computer may be utilised to crack the strongest cryptographic tools now
Indeed, the only cryptographic tools that will survive the era of quantum
computers may be quantum cryptographic tools (whatever that may mean).
Cryptography is, of course, increasingly relevant to law. Cryptoeconomic
innovations –such as the blockchain, cryptocurrencies and smart contracts
–fundamentally rely on robust cryptography to substitute for trust in human
and institutional intermediaries. These promising sets of new social arrange-
ments may be struck down by the unleashed ability to crack encrypted locks.
See discussion of one-way functions in terms of computational complexity in Aaronson (n 14) 101–108.
Certain one-way functions yield when ‘trapdoor’information is provided. ibid.
Shor’s Algorithm was named after Peter W Shor. For a comprehensive introduction to the algorithm,
see Peter W Shor, ‘Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a
Quantum Computer’(1999) 41 Society for Industrial and Applied Mathematics 303.
8J. ATIK AND V. JEUTNER
The discovery of Shor’s Algorithm –and the prospect of quantum computers
that can implement it –speaks to law.
Moreover, Shor’s Algorithm and the surprising quantum solution to a
particular problem that has eluded classical computing suggests a journey
of discovery to identify currently intractable legal problems that may yield
to quantum computing.
The investigation has (at least) two essential steps:
(1) The identiﬁcation of the relevant legal problem that cannot be resolved
by classical techniques but which might be solved by quantum comput-
(2) The speciﬁcation of a quantum algorithm that ‘solves’the legal problem.
Quantum computers, like all computers, represent capacity. To be
eﬀective they require programming. Here the input would be functional ana-
logues to Shor’s Algorithm: stepwise methods using intermediate quantum
states to output a stable solution to the legal problem.
For quantum computing to illuminate (and operationalise) law, law must
be expressed in operational terms. Thus, the ﬁrst order of business would
require creating models of legal processes. Modelling law exposes its under-
lying logic, as well as its indeterminacies. It will permit the judge and lawyer
to see law as problems seeking a solution.
Some of these problems may well be suited to solution by quantum com-
puting. But these solutions will in turn require the discovery of quantum
algorithms. We cannot know for sure the availability of a quantum solution
to a legal question until we ﬁnd it. But we must ﬁrst deﬁne the legal question.
Only if we pose a legal question in computational form can we discover
whether the utilisation of a quantum computer could bring advantages or
whether a classical computer is suﬃciently competent to resolve a given
problem. It is against that background that we now turn to a more detailed
consideration of computational complexity theory.
2.3. Computational complexity theory
Computational complexity theory has developed to capture the state-of-the-
art capacity of classical computers to solve speciﬁc categories of problems.
It roughly divides problems into broad classes, depending on the amount of
time (which reﬂects the number of discrete calculations employed by a par-
ticular algorithm) that a classical computer would need to output a result. As
more factors are involved in a scenario, solutions become more complex.
See the chapter P, NP, and Friends in Aaronson (n 14) for an accessible introduction to computational
complexity theory and the ‘zoology’of P,NP and other complexity classes.
LAW, INNOVATION AND TECHNOLOGY 9
Generally, linear problems are easiest. With respect to linear problems the
number of computer operations increases in proportion to the number of
factors involved. Polynomial functions are more complex. These are problems
where the number of computer operations increases proportionally to a par-
ticular power of the number of factors. Most complex, in this hierarchy, are
exponential functions. Here the number of operations grows by a power pro-
portional to the number of factors. As the number of factors increases, these
problems quickly exceed the capacity of classical computers to solve.
Computational complexity theory rates the particular class of problem by
its best-known algorithm. As such, a ‘hard’problem (one that would over-
whelm the most powerful classical computers) can be rendered ‘easy’(that
is, easily solved by a classical computer) by the discovery of a superior algor-
ithm. As such, today’s‘hard’problems may not necessarily remain so. Math-
ematicians and computer scientists are exploring new algorithmic techniques
and (occasionally or frequently, depending on your view) achieving break-
throughs. At times, these may be startling –such as reducing the best algor-
ithm from an exponential or power function to a polynomial function. At
other times, the progress in reducing complexity may be incremental.
As a science, indeed as a ﬁeld of mathematics, conventional compu-
tational complexity theory is founded upon the operational capacities of clas-
sical computers. Quantum computers bring quite diﬀerent capacities, but
these do not directly map onto the schema of classiﬁcations developed for
computational complexity theory. By engaging computational complexity
theory, we seek to identify vulnerabilities in classical computing rather
than strengths in quantum computing. But as a ﬁrst-order exercise, compu-
tational complexity theory leads us towards discoveries of ﬁelds where
quantum supremacy may lie.
3. Quantum computational law
Having outlined the essential characteristics of quantum computing above,
this section aims to sketch the potential promise of quantum computing for
computational law. In order to do so, we begin by recalling brieﬂy the charac-
ter and operation of computational law (3.1). Subsequently, we explain the
promise of marrying computational law with quantum computing (3.2).
3.1. Classic computational law
Computational law concerns the expression, application and analysis of law in
algorithmic form. It involves forming legal algorithms that proceed through
Aaronson (n 14) at 54 points out that the ultimate determinant of computational complexity is the
availability of a practical algorithm.
10 J. ATIK AND V. JEUTNER
logical processes (at the level of computer hardware, through passage between
‘logic gates’) to create legal conclusions. In this way, computational law
mirrors the ability of classical computers to manipulate arrays of Boolean
states. A piece of Boolean data reﬂects one of two mutually exclusive states,
such as ‘Yes’or ‘No’,or‘True’or ‘False’. Each ‘bit’(for ‘binary integer’)ina
computer records a Boolean alternative, either a ‘0’or a ‘1’. At the level of a
particular bit, computers can only see the Boolean possibilities black or
white: there are no shades of grey. Gradients can be modelled, however,
only by using more bits, digitally mixing proportions of black and white. As
computer operations are carried out, the recorded value of a particular bit
may shuttle from ‘0’to ‘1’and back. In order for law to be translated into a
computational form, at least for classical computers, it must conform to the
digital requirement of a Boolean foundation. While it is debatable whether
all law is Boolean, or binary, in nature,
it is certainly the case that many
legal processes can be viewed as motion from one state to another. For
example, following the logic pathway explored by Gardner,
one can move
from the initial state of ‘no contractual obligation’to the intermediate state
of ‘valid oﬀer’to the ultimate state of ‘contract’.
Traditionally, legal doctrines provide human language accounts of these
changes of states. Computational law abstracts human language renderings of
the process of contract formulation. Such abstract representations may involve
a Boolean logic tree, wherein the presence or absence of certain factors (Yes or
No responses to a Boolean query) moves the process through a logical structure
that will (or will not) reach a conclusion that constitutes a change of state.
Computational law is a new ﬁeld, but it is already evident that much law
will be expressed in machine readable code, instead of human language. In
most instances, legal reasoning is quite simple. That is, once the translation
from human language to machine readable code is completed, there is little
to computationally challenge a common computer.
However, for our purposes the question is whether and how the emer-
gence of quantum computing could actually enhance the powers of classical
computational law. We explore this question in the next section below.
3.2. Quantum-powered computational law
The emergence of quantum computers entails the potential to signiﬁcantly
enhance the powers of computational law. One of the clearest ways to
For a prominent theory of the binary character of legal systems, see Niklas Luhmann, Law as a Social
System (Fatima Kastner and others eds, Klaus A Ziegert tr, Oxford Socio-Legal Studies, 2004). For recent
critiques of this view, see Rostam J Neuwirth, Law in the Time of Oxymora: A Synaesthesia of Language,
Logic and Law (Routledge, 2018); Jeutner (n 24).
Anne von der Lieth Gardner, An Artiﬁcial Intelligence Approach to Legal Reasoning (MIT Press, 1987).
LAW, INNOVATION AND TECHNOLOGY 11
illustrate the potential of quantum-powered computational law relates to the
(in)famous debate concerning law’s (in)deterministic character. One of the
central claims of the Critical Legal Studies movement, and a claim frequently
discussed across many sub-ﬁelds of legal philosophy,
is that law is inher-
Computational law presumes just the opposite: that
much of law is deterministic and can be faithfully expressed in algorithmic
form. Given certain inputs and the execution of the algorithm, a consistent
outcome is expected to result. Computational law that engages quantum
computing may reconcile these two views, permitting robust outputs while
addressing the well-recognised sources for law’s asserted indeterminacy.
Classical computing is certainly deterministic. There is no algorithm that
will produce one result one time and another result the next. Rather, classical
computation is repeatable. If law as we now know it is imperfectly determi-
nistic, then translating legal operations into computational form will result in
imposing determinism on it, which would constitute a fundamental change
to the law we have known.
But imagine the Critical Legal Studies writers have the nature of law right.
What does it mean to say that law is indeterminate? Legal writers have
noticed the simultaneous availability of contrarieties –the presence
of conﬁrmed and vital doctrines that lead to inconsistent results.
trine is fully deterministic. As such, the initial selection of which doctrine to
apply largely determines the outcome. In the realist view of law, the judge
subconsciously chose that particular doctrinal alternative (from the twin
In some respects, there is a remarkable similarity between the discussions concerning law’s (in)deter-
minacy and the debate among natural scientists concerning the (in)deterministic nature of the physical
world Compare, for example, Pierre Simon Laplace, A Philosophical Essay on Probabilities (Frederick
Wilson Truscott and Frederick Lincoln Emory trs, Chapman & Hall, 1902) 4; Albert Einstein, Boris
Podolsky and Nathan Rosen, ‘Can Quantum-Mechanical Description of Physical Reality Be Considered
Complete?’(1935) 47 Physical Review 777, 780; Albert Einstein, ‘Physik Und Realität’(1936) 221 Journal
of the Franklin Institute 313, 342–343; Albert Einstein, Max Born and Hedwig Born, Briefwechsel 1916–
1955 (Rowohlt 1972) 98; Niels Bohr, Atomic Theory and the Description of Nature (Cambridge University
Press, 1934) 109; Niels Bohr, ‘Discussion With Einstein on Epistemological Problems in Atomic Physics’
in J Kalckar (ed), Foundations of Quantum Physics II (1933–1958) (Elsevier, 1969); Stephen Hawking,
‘Does God Play Dice?’<https://www.hawking.org.uk/in-words/lectures/does-god-play-dice> accessed
21 January 2021.
See, for example, Roberto Mangabeira Unger, ‘The Critical Legal Studies Movement’(1983) 96 Harvard
Law Review 561; Charles M Yablon, ‘The Indeterminacy of the Law: Critical Legal Studies and the
Problem of Legal Explanation’(1984) 6 Cardozo Law Review 917; Lawrence B Solum, ‘On the Indeter-
minacy Crisis: Critiquing Critical Dogma’(1987) 54 University of Chicago Law Review 462.
Richard Nobles and David Schiﬀ, for instance, note that ‘legal statements …oscillate between contra-
dictions that cannot be accounted for through further reﬁnement’, Richard Nobles and David Schiﬀ,
‘Review of Paradoxes and Inconsistencies in the Law by Oren Perez and Gunther Teubner (eds)’
(2007) 70 Modern Law Review 505, 509. Similarly, Singer observers that ‘sometimes the best way to
express our values and our social and legal practices is by adopting what seem to be contradictory
principles, even though we cannot now, and perhaps never will, be able to reconcile them fully.’
Joseph William Singer, Entitlement: The Paradoxes of Property (Yale University Press 2000) 204–205.
For related observations concerning international law, see Martti Koskenniemi, From Apology to
Utopia: The Structure of International Legal Argument (Cambridge University Press 2005) 65.
12 J. ATIK AND V. JEUTNER
set of contrarieties) to reach an outcome desired for other, largely unprin-
It may be that the observed presence of contrarieties throughout law –
with the resultant facility of an algorithm to output diﬀerent results –can
be better modelled using a quantum computer. Quantum computing incor-
porates fundamental quantum attributes, such as superposition. In quantum
physics, a superposition exists when the state of a particle (an electron, for
example) is suspended between two physical states. As a result, the particle
occupies, at least apparently, those two states at the same time. The most
famous illustration of such a superposition state involves a cat in a box
that is suspended between the state of being dead or alive.
computers this phenomenon is utilised to enable signiﬁcantly faster proces-
sing of information. Strictly speaking, a superposition state is not the same as
indeterminacy. Nonetheless, in law, this phenomenon could be helpful to
conceptualise situations in which a certain kind of conduct is, as a matter
of positive law, both legal and illegal at the same time. For example, when
a contract or a treaty prohibits an act while also making it obligatory, or
when norms belonging to diﬀerent normative orders (for example, domestic
and international law) collide. Those situations occur rarely (and they are for
law an anomaly of the same magnitude as quantum physics is for classical
physics), but when they do occur references to the quantum physical super-
position phenomenon could help lawyers making sense of such otherwise
merely irresolvable norm conﬂicts.
Similar questions arise with respect to law’s rule / exception dynamic –a
phenomenon that has been well explored in computational legal theory.
Classical computing more than adequately handles the rule / exception
dynamic once the rule and any accompanying exception have been
deﬁned. But classical computing is inadequate in predicting when a new
exception will be found. The formation of a new exception constitutes a
rupture from the settled deterministic pathway. Here the phenomenon of
superposition may serve to better model this kind of legal phenomenon.
Erwin Schrödinger, ‘Die gegenwärtige Situation in der Quantenmechanik’(1935) 23 Naturwissenschaf-
ten 807, 812. See also Serge Haroche and Jean-Michel Raimond, Exploring the Quantum: Atoms, Cav-
ities, and Photons (Oxford University Press, 2006) 19, 27, 70–71, 82.
See generally, Jeutner (n 24). Ted Sichelman, in his work on the use of quantum game theory to model
the intellectual property regime, suggests that IP rights can be better described probabilistically. IP
rights lie somewhere on a continuum between the poles representing the absence of IP rights and
‘ironclad’IP rights that provide the holder with absolute protection –a metaphorical form of
quantum superposition, Ted M Sichelman, ‘Quantum Game Theory and Coordination in Intellectual
Property’(Social Science Research Network 2015) SSRN Scholarly Paper ID 1656625 <https://papers.
ssrn.com/abstract=1656625> accessed 20 June 2021.
For a contemporary treatment of this topic, see Luís Duarte d’Almeida, Allowing for Exceptions: A Theory
of Defences and Defeasibility in Law (Oxford University Press, 2015). See also, Glanville Williams, ‘The
Logic of “Exceptions”’ (1988) 47 Cambridge Law Journal 261; Kevin D Ashley (n 12) 58–59; Federica
Paddeu and Lorand Bartels (eds), Exceptions and Defences in International Law (Oxford University
LAW, INNOVATION AND TECHNOLOGY 13
Indeterminacy also results from the arbitrary level of abstraction that
underlies most legal reasoning. There are few legal formulas emptier than
‘decide like cases alike’. At one extreme, there are no like cases. There is an inter-
mediate level of discrimination thatmakes law work: law does assume the exist-
ence of some like cases. But diﬀerent legal systems display diﬀerent levels of
discrimination: two cases may be alike in one system, but subject to disparate
outcomes in another. The same applies within a legal system over time; two
cases may be ‘like’in one period, yet unlike in another. The level of abstraction
can move from lesser to greater magniﬁcation –and back. There is no determi-
nate legal principle that anchors the coarseness or ﬁneness of the categorisation
of cases. Here too, quantum computing may open up the possibility for better
(that is, more insightful) modelling of this complex legal phenomenon.
Finally, indeterminacy results from the uncertainty of the quantity and
quality of information that serves as inputs to a legal process (or algorithm).
Legal institutions are constitutionally starved of information. Only a minuscule
fraction of the context in which a legal question arisesis harvested and conveyed
to a tribunal. More or less information can drastically swing outcomes. Classical
law compensates for these limitations, at times, by applying presumptions for
certain contextual predicates –but the very application of these presumptions
–highly uncertain for the most part –introduces new indeterminacy at the
same time as it eliminates other indeterminacy. Computational law introduces
the prospect for a much fuller provision of information, and hence greater accu-
racy of results.Yet at the granular level, more data creates more computational
complexity. This in turn may render classical legal algorithms which function
reliably in a data-deprived setting to no longer work. Accordingly, companies
and public authoritiesare alreadyexperimenting with the use of quantumcom-
puters due to their inherent ability to consider vastly more factors in a single
computational stroke than classical computing.
For example, biomedical companies are experimenting with the use of
quantum computers to enhance radiotherapy by simulating thousands of
variables in order to devise radiation plans that avoid damaging healthy
Quantum computing could also be used to enhance the AI involved
in machine learning and pattern recognition to optimise patient diagnos-
Moreover, car manufacturers are joining forces with quantum tech-
nology companies to improve traﬃcﬂow predictions in an attempt to
reduce congestion and optimise travel time. Public service providers use
quantum computing to model and optimise the provision of electricity
and water. In space, quantum computing is utilised to control the movement
and quantum computing can also be used to optimise AI
Roswell Park Cancer Institute, ‘Quantum Annealing Applied to Optimization Problems in Radiation
Medicine’<https://www.dwavesys.com> accessed 16 January 2019.
See generally, ‘Quantum Computing Set to Revolutionise the Health Sector’(L’Atelier BNP Paribas)
<https://atelier.bnpparibas> accessed 21 June 2021.
14 J. ATIK AND V. JEUTNER
predicting the outcome of general elections.
Thus, the advent of quantum
computing promises to remedy contemporary legal challenges related to the
uncertainty of the quantity and quality of information by equipping compu-
tational law to deal much better with signiﬁcantly larger data sets.
Overall, quantum computing and its phenomenon of superposition allows
computational law to capture and produce more muted, more subtle legal
outputs. This possibility –together with increasing command in generated
quantum legal algorithms –may not only facilitate the making of new law
but might also shed new light on the debate concerning law’s (in)determi-
nacy. Going beyond these more foundational observations, the next
section continues to consider the potential of quantum computers for
three concrete legal processes.
4. Targets for quantum legal algorithms
Against the background of the more foundational discussion of the potential
promises of quantum computational law above, this section considers three
more concrete legal ﬁelds of application for quantum computing
concerning optimisation problems (4.1), burdens of proof (4.2) and
machine learning (4.3).
4.1. Optimisation problems
Quantum computing oﬀers much promise for the solution of multifactor
optimisation problems. Optimisation mathematics is a well explored
feature of industrial economics. A steel plant may use a variety of inputs,
such as coal, iron, and labour, each with its respective cost per unit. One
or more factors may be subject to a ﬁnite bound. Given the mix of inputs,
an optimisation analysis involves maximising the proﬁt from steel manufac-
ture, which involves ﬁnding a solution that proﬁtably maximises the output
of steel and minimises the costs of the available inputs. Such optimisation
problems are tractable when the number of factors (output of steel and
coal, iron and labour inputs) are few in number. When the number of
factors expands, determining the optimal mix involves an ever-increasing
amount of computation. Optimisation problems become signiﬁcantly
more complex when some factors are partial or total substitutes for others
or where the ratio of factors employed can vary. Moreover, additional com-
plexity can result from shifts in prices for the ﬁnished good or in the costs of
Gideon Bass and Booz / Allen / Hamilton, ‘Heterogeneous Quantum Computing for Satellite Optimiz-
ation’(September 2017) <https://www.dwavesys.com>.
Max Henderson, ‘Quantum Machine Learning for Election Modeling’<https://www.dwavesys.com>
accessed 21 June 2021.
LAW, INNOVATION AND TECHNOLOGY 15
the inputs. Quantum computing may permit far more accurate solutions to
determining the proper mix.
Law often poses optimisation challenges. Every time a judge invokes a
‘balancing test’, she is asking herself to optimise some output (quanta of
justice, perhaps) given the imposition and distribution of legal burdens on
the parties before her. Consider the four-factor test applied by an American
judge (exercising her equitable discretion) to determine whether to grant a
.plaintiﬀwould suﬀer an irreparable injury
.inadequate remedy at law
.balance of hardships
In eBay v. MercExchange,
the US Supreme Court presents this four-
factor test as an ordinary judicial operation.
To translate this determination
into an algorithm reveals layers of computational complexity. The ﬁrst factor
–whether the plaintiﬀwould suﬀer irreparable injury –seems the easiest to
formulate. It takes a Boolean form: the judge is asked to give a ‘Yes’or ‘No’
response. But what constitutes, as a matter of law, an irreparable injury may
itself be a complex determination involving many factors.
The second factor –whether an adequate remedy is available at law –is
familiar to common law lawyers. This does not mean, however, that it
involves a simple calculation. While in many situations, the relevant prospec-
tive remedy at law is damages, the doctrine admits the possible presence of
other remedies. But even if we conﬁne the exercise to an assessment of the
adequacy of damages to remedy a harmed plaintiﬀ, there is more complexity
here than might ﬁrst meet the eye.
The third factor –the balance of hardships between plaintiﬀand defen-
dant –appears to be a fairly simple calculation, involving the quantitative
measurement of the harm to the plaintiﬀthat would result in the absence
of injunctive relief and the harm caused to the defendant should that relief
be granted. The algorithm here asks if the ﬁrst element exceeds the
second. Again, what might appear at ﬁrst to be relatively simple may turn
out to be complex. The heart of the computation is the measurement of
the two harms in play. Both involve forward looking speculation. And
each has at least some element of assessing a counterfactual situation,
depending on the duration of the behaviour sought to be enjoined. More-
over, the third factor constitutes an inequality expression. Mathematically
inequality relationships greatly increase the complexity of a relationship.
eBay, Inc v MercExchange, LLC, 547 US 388 (2006).
The eBay Court describes this four factor test as ‘well-established’and ‘traditional’.
16 J. ATIK AND V. JEUTNER
The fourth factor –consideration of the public interest –is likely beyond
translation into computational form. The various aﬀronts to the public inter-
est that might justify a refusal to grant an otherwise meritorious injunction is
limited only by the huge number of cases where such public interest con-
siderations have been evaluated.
Here are two more multifactor tests familiar to technology lawyers: the
four non-exclusive factors set out in Section 107 of the US Copyright Act
for assessing fair use
and the mind-boggling 15-factor test for determining
reasonable patent royalty rates enunciated in Georgia Paciﬁc.
Both of these
Section 107 of the Copyright Act of 1976 commands a judge considering an assertion of ‘fair use’to
(1) the purpose and character of the use, including whether such use is of a commercial nature or is
for nonproﬁt educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole;
(4) the eﬀect of the use upon the potential market for or value of the copyrighted work.
17 USC §107.
See Georgia-Paciﬁc v US Plywood Corp, 318 F SUPP 1116, 1120 (SDNY 1970) for a list of factors to be
considered by a court in ﬁxing a reasonable patent royalty:
1. The royalties received by the patentee for the licensing of the patent in suit, proving or tending to
prove an established royalty.
2. The rates paid by the licensee for the use of other patents comparable to the patent in suit.
3. The nature and scope of the license, as exclusive or non-exclusive; or as restricted or non-
restricted in terms of territory or with respect to whom the manufactured product may be sold.
4. The licensor’s established policy and marketing program to maintain his patent monopoly by not
licensing others to use the invention or by granting licenses under special conditions designed to
preserve that monopoly.
5. The commercial relationship between the licensor and licensee, such as, whether they are com-
petitors in the same territory in the same line of business; or whether they are inventor and
6. The eﬀect of selling the patented specialty in promoting sales of other products of the licensee;
the existing value of the invention to the licensor as a generator of sales of his non-patented
items; and the extent of such derivative or convoyed sales.
7. The duration of the patent and the term of the license.
8. The established proﬁtability of the product made under the patent; its commercial success; and its
9. The utility and advantages of the patent property over the old modes or devices, if any, that had
been used for working out similar results.
10. The nature of the patented invention; the character of the commercial embodiment of it as
owned and produced by the licensor; and the beneﬁts to those who have used the invention.
11. The extent to which the infringer has made use of the invention; and any evidence probative of
the value of that use.
12. The portion of the proﬁt or of the selling price that may be customary in the particular business or
in comparable businesses to allow for the use of the invention or analogous inventions.
13. The portion of the realisable proﬁt that should be credited to the invention as distinguished from
non-patented elements, the manufacturing process, business risks, or signiﬁcant features or
improvements added by the infringer.
14. The opinion testimony of qualiﬁed experts.
15. The amount that a licensor (such as the patentee) and a licensee (such as the infringer) would
have agreed upon (at the time the infringement began) if both had been reasonably and volun-
tarily trying to reach an agreement; that is, the amount which a prudent licensee who desired, as a
business proposition, to obtain a license to manufacture and sell a particular article embodying
LAW, INNOVATION AND TECHNOLOGY 17
formulations pack more complexity than a mere mortal judge can possibly
disentangle. It is conceivable that a quantum computer could process a
complex algorithm that pays more than judicial lip-service to these factors
and their respective interdependencies.
4.2. Burdens of proof
Much of the algorithmic structure of law takes the form of simple conditional
statements. If various factors are present (that is, if they take a Boolean
value), then a legal conclusion results or a change in legal state is eﬀected.
If only law were so simple. The conditions to be satisﬁed are frequently
quite complex themselves. Whether a valid oﬀer has been presented is a con-
dition that must be satisﬁed in order to conclude that a valid contract has
However, this condition itself constitutes a legal conclusion
that algorithmically results in the satisfaction of its own set of conditions.
Legal algorithms cascade backwards, many inputs are themselves outputs
of logically prior algorithmic operations.
But there is more detail to be found within many legal algorithms. The
simple form of many algorithms involves a grammar of conditionality. In
order to determine whether a particular input is indeed present or not,
lawyers require the satisfaction of a burden of proof. The presence or
absence of a particular input may be thought of as a continuum, rather
than a mutually exclusive, bipolar distribution that signals the presence or
absence of a basis for a legal conclusion.
Some legal factors are necessary in the sense that a positive outcome
cannot result in the absence of those factors. Other factors can have positive
weight and can contribute (in a mix with other factors) to a legal outcome,
but their absence does not compel a negative result. The interplay of necess-
ary and contributory factors is more complex than would be the case were
each factor independent and necessary.
Law is replete with burdens of proof.
Burdens of proof are legal con-
structs that operate like step functions. Up to a certain point, the probabil-
istic presence of factors (necessary or contributory) do not throw oﬀa
change in legal state. But there is a point when one says the burden has
been satisﬁed where the legal conclusion results. Precisely where this critical
point lies is not known –but it is the conventional understanding that the
point exists and that the judge (who, at least in the Anglo-American legal
the patented invention would have been willing to pay as a royalty and yet be able to make a
reasonable proﬁt and which amount would have been acceptable by a prudent patentee who
was willing to grant a license.
See Gardner (n 33).
See the discussion of ‘proof standards’in computational models of legal arguments presented in Kevin
D Ashley (n 12) 145–146.
18 J. ATIK AND V. JEUTNER
system, assesses whether the burden of proof is satisﬁed) is able to identify it
in making a particular legal determination. A broader estimate of the pos-
ition of the critical point is obtained by collecting positive and negative
examples of satisfaction of the burden of proof within a particular category
of cases. This estimate serves both the lawyers (in their estimation of the
strengths and weaknesses of their respective cases) and the judge in
making further burden of proof determinations.
In some sense, recourse to burdens of proofs permits the judge to reach
backward into the cascade of preliminary conclusions while facing the
ultimate legal conclusion that must be rendered. Consider the ultimate
conclusion facing the judge or jury in a criminal proceeding: whether
or not the defendant has committed the oﬀence. Each deﬁned crime has
elements that function to deﬁne an algorithm leading to the imposition
of criminal responsibility. But these elements are themselves complex
and escape simple speciﬁcation (what does constitute ‘malice
As a general matter, all of the elements of a crime are essential factors, but
a judge or jury may reach the ultimate conclusion of guilt or innocence by a
mysterious recourse to the high burden of proof applied in criminal cases:
beyond a reasonable doubt.
What would a legal algorithm look like that respected both the deﬁned
elements of the crime (and the backward cascade of lower-level elements
supporting each) as well as an overall burden of proof?
Burdens of proof can also be thought of as involving mathematical
inequalities that relate all subsidiary elements. Consider the common law
crime of burglary. In its simple form, the crime has two elements: entering
a structure illegally and having the intent to commit a crime.
would depend on the presence of (1) illegal entry and (2) intent to commit a
crime. Both of these factors would have to be present in order to conclude
that a burglary took place. Each of these elements is itself a complex legal
conclusion: what constitutes an illegal entry and what constitutes intent to
commit a crime. To a large degree the two factors are interrelated: the
intent requirement refers both to the entry (in that it motivates the entry)
and to the goal of the entry. Lower level factors that suggest illegal entry
(the presence of certain tools) may also contribute to a demonstration of
intent. Here too a quantum computer may be able to fully capture the
nuanced interplay of multitudinous subsidiary factors that comprise
elements of the legal algorithm.
Burglary is deﬁned at common law as ‘[t]he breaking and entering the house of another in the night
time, with intent to commit a felony therein, whether the felony be actually committed or not.’
LAW, INNOVATION AND TECHNOLOGY 19
4.3. Machine learning and quantum computational law
Machine learning will be intensively utilised in many computational law
applications. The legal system presents a complex of rules, principles and per-
mitted and forbidden lines of reasoning. The law is diﬃcult enough where its
sources are conﬁned to statutes and secondary legislation; the task of model-
ling a legal system is much more diﬃcult where formal status is assigned to
decisional law, as in the Anglo-American common law system. Judicial
decisions state and re-state legal notions. Generally, but not always, these
decisions follow validated reasoning pathways. The demand to recognise
what constitutes law –for the purpose of building a model –likely exceeds
the capacity of any human lawyer. Machine learning promises the ability to
access vastly greater loads of data and to ﬁnd new and old patterns within
them. In the case of law, machine learning may be more able to ‘restate’
what the law is by including in its base a vastly greater number of decisions.
Machine learning has been identiﬁed as an area that may beneﬁt from
quantum computing. Machines ‘learn’by adjusting the weights given to
various input factors to create better and better predictions (that is,
outputs that can be evaluated and validated). Every exercise of a machine
is an opportunity to reﬁne its power and accuracy, through small adjust-
ments to the weights assigned to the various linkages that comprise the artiﬁ-
cial neural network. These slight adjustments are accepted when they lead to
better results; otherwise the direction of adjustment is reversed. Through a
process known as ‘gradient descent’a learning machine seeks to improve
the power of its appreciation. The artiﬁcial neural network can be rep-
resented by a system of linear equations. Together, these may generate a
function that includes local maxima and minima (peaks and valleys). A
single highest point can be found by a learning machine by divining upslop-
ing and downsloping moves. A problem results where there is more than one
peak. Here, gradient descent may bring the machine to a local peak beyond
which there is another valley. The learning machine cannot feel its way to the
even higher peak on the other side of that valley. Which is to say, it will not
reach the optimal values in every case.
Quantum computing promises to solve this problem –as it will permit the
machine to see its way beyond its current locale. Finding the optimal model
will become a more certain process if quantum computing can be deployed.
If quantum computing can bring about a general enhancement to
machine learning, computational law will certainly beneﬁt. Learning
machines eﬀectively write algorithms (though these algorithms may not be
transparent). If one wanted to say what the law is in a particular domain –
based on the entire corpus of relevant cases –machine learning will likely
See Introduction in Michael A Livermore, Law as Data: Computation, Text, and the Future of Legal Analy-
sis (Daniel N Rockmore ed, SFI Press, 2019).
20 J. ATIK AND V. JEUTNER
outperform the best drafting committee of the American Law Institute.
Quantum computing will enhance the accuracy of the resultant model by
locating the relevant optimal mix of values and weights.
The anticipated beneﬁts of quantum computing for the natural sciences and for
the global technology sector are frequently commented upon. This essay aimed
to show that quantum computing is also a legally relevant phenomenon.
established above, quantum computing technology could be utilised to greatly
enhance the abilities of the emerging ﬁeld of computational law to express,
apply and analyse law in algorithmic form. In order to advance our thesis,
we referred to certain quantum phenomena, such as superposition states,
and drew parallels between normative discussions concerning (in)determin-
ism in the natural science and in legal theory. Indeed, we share the view that
‘the metaphors and intuitions that guide physicists can enrich our comprehen-
sion of social and legal issues’
and that reﬂection ‘upon certain developments
in physics can help us hold on to and reﬁne some of our deeper insights into the
pervasive and profound role law plays in shaping our society’.
At ﬁrst sight, these attempts to invoke and refer to quantum mechanical
phenomena in a legal context might appear to be peculiar. Historically,
however, it is by no means unusual that developments in the natural sciences,
in general, and in physics, in particular, generate knock-on eﬀects across the
(perceived) natural / social science divide.
In the legal sphere, the emer-
gence of computational law and legal analytics is itself an example of the
fusion of ordinarily separate strands of mathematical and legal reasoning.
It is against that background that this essay has argued that legal academics
and practicing lawyers alike should reﬂect upon the legal signiﬁcance of the
unfolding ‘Second Quantum Revolution’.
In order to facilitate serious investigations into the legal utility of quantum
computing technology, we argued that two conceptual steps need to be taken:
ﬁrst, it must be identiﬁed with respect to which kind of problems quantum
computers enjoy ‘quantum supremacy’and, second, the contours of
For thoughts on the legal and social dimension of quantum computers, see Valentin Jeutner, ‘The
Quantum Imperative: Addressing the Legal Dimension of Quantum Computers’(2021) 1 Morals &
Tribe (n 25) 2.
In the context of history, for example, Clark observes that the debate of physicists concerning uncer-
tainty and indeterminacy and ‘their claim that the position of the observer must be taken into account
in measuring space and time, cast doubt on the model of the omniscient observer assumed by classical
theories of knowledge’at the beginning of the 20th century. Elizabeth A Clark, History, Theory, Text:
Historians and the Linguistic Turn (Harvard University Press, 2004) 16. See also, Alexander Wendt,
Quantum Mind and Social Science (Cambridge University Press, 2015).
LAW, INNOVATION AND TECHNOLOGY 21
‘quantum supremacy’thus identiﬁed need to be related to the legal context.
With respect to the ﬁrst step, we sought to explain, with reference to compu-
tational complexity theory, how the superiority of quantum computers
as compared to classical computers relates, in principle, to problems of
NP-nature. That is, quantum computers are superior only with respect to
non-deterministic problems for which no known, practical algorithm exists
that can shorten the pathway to an eﬃcient solution. With respect to the
second step, we then demarcated the possible contours of legal ‘quantum
supremacy’by showcasing three anticipated legal ﬁelds of quantum technol-
ogy: optimisation problems, burdens of proof, and machine learning. These
three use-cases are only examples of potential applications of quantum com-
puting technology in the legal sphere. There might be many more. At the same
time, however, it should also have become apparent that the legal signiﬁcance
of quantum computers will, in any event, be limited to a speciﬁc subset of legal
questions. Indeed, we explicitly acknowledge that there are many legal issues
that (quantum) computational law cannot adequately capture.
Even with respect to the subset of legal issues that could be aﬀected by the
emergence of quantum computing technology, the exact manifestation of
quantum computational law remains, at this point in time, speculative
since the successful application of quantum computing to the ﬁeld of law
requires the concurrent evolution of two uncertain lines of development:
on the one hand, quantum computers that can work on an eﬃcient scale,
and, on the other hand, a more advanced ability to translate law from
human language to computer code. Only a convergence of these two lines
of development will ring in the era of computational law.
When these two lines of development converge is diﬃcult to predict. But
despite these uncertainties, we submit that there is merit in embarking upon
this essay’s unapologetic consideration of the legal signiﬁcance of quantum
technology in the future since the meaningful utilisation of quantum com-
puting in the legal sector at a later stage presupposes a creative imagination
of possible use-cases at the present.
The authors have explored quantum theory and law with Karl Manheim and Timo
Minssen, fellow collaborators within the Quantum Law Project at Lund. They are
grateful for comments and suggestions from Elizabeth Pollman and Ted Sichelman.
The authors enjoyed the support of colleagues at Lund and Loyola, including Xavier
Groussot, Justin Levitt, Mia Rönnmar, Michael Waterstone and Lauren Willis. Por-
tions of this essay were presented at the Second International Computational Law
Forum held at Tsinghua University School of Law, Beijing, September 21 & 22,
2019. The Quantum Law Project is funded by the Wallenberg Programme on AI,
Autonomous Systems and Software –Humanities and Society (WASP-HS).
22 J. ATIK AND V. JEUTNER
No potential conﬂict of interest was reported by the author(s).
Notes on contributors
Jeﬀery Atik is Professor of Law at Loyola Law School in Los Angeles and Guest Pro-
fessor of Civil Law at Lund University. His research focuses on law and artiﬁcial
intelligence, including computational law and national security / international
Valentin Jeutner is Associate Professor of Law at Lund University, Sweden. His
research focuses on foundational questions of (international) law. At Lund, he
serves as the PI of Sweden’s Quantum Law Project.
LAW, INNOVATION AND TECHNOLOGY 23